You put the skewer together correctly. The conical springs point inward like this [>====<]. Also the wobble in the freewheel or cassette is normal, except you never noticed it before.

I suspect that you're hearing a bit of drivetrain noise because the wheel isn't in the identical position as last time. First let's get the wheel back to the natural position. Hold the bike vertical on the ground, open the QR and tap or jiggle the wheel a bit and let it settle home by gravity. The rim should now be centered in the chainstays and seatstays (this doesn't work if you have horizontal dropouts without micro-adjust screws, in which case you have to center the rim in the chainstays by eye). If the rim is not also centered in the brakes, they get centered to the rim, not vice versa. from here on you'll always mount the wheel by gravity and it'll return to the exact same place on it's own.

Now spin the wheel and listen for any rubbing, which will usually be intermittent as the wobbly rim touches the brakes here and there. Adjust or center the brakes to clear until the rub is resolved. Any other sound? If not ride it and listen both coasting, pedaling on the level and also pedaling on a hill. If it makes noise, only while pedaling, try to figure if it's a metallic chain/sprocket noise, or possibly the rim or tire rubbing (sometimes pedaling under load deflects things slightly so a wheel that wasn't rubbing now does). If it's chain noise bit your shifting is 100% fine, ignore it, otherwise check the trim of the RD and fine tune it with the cable adjuster.